That is not good, 0.01 inch is like a mile, way too much. Time to do a bit more research. Is the problem a loose quill or is it the spindle?
Would put a long(ish) bar in the spindle where you can get some hand leverage on it and test both the quill and spindle nose to see which is moving. Then take corrective action. Hope that the problem is the spindle bearings, a loose quill is a major problem. But I have never heard of machine having a quill that loose.